Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties

Caenepeel, S.; Militaru, G.; Zhu, Shenglin

doi:10.1090/S0002-9947-97-02004-7

Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties
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by S. Caenepeel, G. Militaru and Shenglin Zhu PDF

Trans. Amer. Math. Soc. 349 (1997), 4311-4342 Request permission

Abstract:

We study the following question: when is the right adjoint of the forgetful functor from the category of $(H,A,C)$-Doi-Hopf modules to the category of $A$-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that $C\otimes A$ and the smash product $A\# C^*$ are isomorphic as $(A, A\# C^*)$-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case $A=H$, and this leads to the notion of $k$-Frobenius $H$-module coalgebra. In the special case of Yetter-Drinfel′d modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if $H$ is finite dimensional and unimodular.

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